For the Dirichlet forms \( (\mathcal {E} ^1, \mathcal {D}^1 )\)  and \( (\mathcal {E} ^2, \mathcal {D}^2 )\) , we write
\( (\mathcal {E} ^1, \mathcal {D}^1 ) \le (\mathcal {E} ^2, \mathcal {D}^2 ) \)  if
\( \mathcal {E} ^1 (f,f) \le \mathcal {E} ^2 (f,f) \)  for all \(f \in \mathcal {D}^2\)  and \( \mathcal {D}^1 \supset \mathcal {D}^2 \) .
We say that \( \lbrace  (\mathcal {E} ^i, \mathcal {D}^i) \rbrace _{i} \)  is increasing if
\( (\mathcal {E} ^i, \mathcal {D}^i) \le (\mathcal {E} ^{i+1}, \mathcal {D}^{i+1})\)  for all \( i \) .
We now quote a result from [1]}.
